SOCP based variance free Dantzig Selector with application to robust estimation

Sparse estimation methods based on ℓ ₁ relaxation, such as Lasso and Dantzig Selector, are powerful tools for estimating high dimensional linear models. However, in order to properly tune these methods, the variance of the noise is often used. In this paper, we propose a new approach to the joint estimation of the sparse vector and the noise variance in a high dimensional linear regression. The method is closely related to the maximum a posteriori estimation and has the attractive feature of being computable by solving a simple second-order cone program (SOCP). We establish nonasymptotic sharp risk bounds for the proposed estimator and show how it can be applied in the problem of robust estimation.